# Module Bounds

Computation of resource bounds for Linear code.

Require Import Coqlib.
Require Import AST.
Require Import Op.
Require Import Locations.
Require Import Linear.
Require Import Conventions.

# Resource bounds for a function

The bounds record capture how many local and outgoing stack slots and callee-save registers are used by a function.

We demand that all bounds are positive or null. These properties are used later to reason about the layout of the activation record.

Record bounds : Type := mkbounds {
bound_local: Z;
bound_int_callee_save: Z;
bound_float_callee_save: Z;
bound_outgoing: Z;
bound_stack_data: Z;
bound_local_pos: bound_local >= 0;
bound_int_callee_save_pos: bound_int_callee_save >= 0;
bound_float_callee_save_pos: bound_float_callee_save >= 0;
bound_outgoing_pos: bound_outgoing >= 0;
bound_stack_data_pos: bound_stack_data >= 0
}.

The following predicates define the correctness of a set of bounds for the code of a function.

Section WITHIN_BOUNDS.

Variable b: bounds.

Definition mreg_within_bounds (r: mreg) :=
index_int_callee_save r < bound_int_callee_save b
/\ index_float_callee_save r < bound_float_callee_save b.

Definition slot_within_bounds (sl: slot) (ofs: Z) (ty: typ) :=
match sl with
| Local => ofs + typesize ty <= bound_local b
| Outgoing => ofs + typesize ty <= bound_outgoing b
| Incoming => True
end.

Definition instr_within_bounds (i: instruction) :=
match i with
| Lgetstack sl ofs ty r => slot_within_bounds sl ofs ty /\ mreg_within_bounds r
| Lsetstack r sl ofs ty => slot_within_bounds sl ofs ty
| Lop op args res => mreg_within_bounds res
| Lcall sig ros => size_arguments sig <= bound_outgoing b
| Lbuiltin ef args res =>
(forall r, In r (params_of_builtin_res res) \/ In r (destroyed_by_builtin ef) -> mreg_within_bounds r)
/\ (forall sl ofs ty, In (S sl ofs ty) (params_of_builtin_args args) -> slot_within_bounds sl ofs ty)
| _ => True
end.

End WITHIN_BOUNDS.

Definition function_within_bounds (f: function) (b: bounds) : Prop :=
forall instr, In instr f.(fn_code) -> instr_within_bounds b instr.

# Inference of resource bounds for a function

The resource bounds for a function are computed by a linear scan of its instructions.

Section BOUNDS.

Variable f: function.

In the proof of the Stacking pass, we only need to bound the registers written by an instruction. Therefore, this function returns these registers, ignoring registers used only as arguments.

Definition regs_of_instr (i: instruction) : list mreg :=
match i with
| Lgetstack sl ofs ty r => r :: nil
| Lsetstack r sl ofs ty => r :: nil
| Lop op args res => res :: nil
| Lstore chunk addr args src => nil
| Lcall sig ros => nil
| Ltailcall sig ros => nil
| Lbuiltin ef args res => params_of_builtin_res res ++ destroyed_by_builtin ef
| Llabel lbl => nil
| Lgoto lbl => nil
| Lcond cond args lbl => nil
| Ljumptable arg tbl => nil
| Lreturn => nil
end.

Fixpoint slots_of_locs (l: list loc) : list (slot * Z * typ) :=
match l with
| nil => nil
| S sl ofs ty :: l' => (sl, ofs, ty) :: slots_of_locs l'
| R r :: l' => slots_of_locs l'
end.

Definition slots_of_instr (i: instruction) : list (slot * Z * typ) :=
match i with
| Lgetstack sl ofs ty r => (sl, ofs, ty) :: nil
| Lsetstack r sl ofs ty => (sl, ofs, ty) :: nil
| Lbuiltin ef args res => slots_of_locs (params_of_builtin_args args)
| _ => nil
end.

Definition max_over_list {A: Type} (valu: A -> Z) (l: list A) : Z :=
List.fold_left (fun m l => Zmax m (valu l)) l 0.

Definition max_over_instrs (valu: instruction -> Z) : Z :=
max_over_list valu f.(fn_code).

Definition max_over_regs_of_instr (valu: mreg -> Z) (i: instruction) : Z :=
max_over_list valu (regs_of_instr i).

Definition max_over_slots_of_instr (valu: slot * Z * typ -> Z) (i: instruction) : Z :=
max_over_list valu (slots_of_instr i).

Definition max_over_regs_of_funct (valu: mreg -> Z) : Z :=
max_over_instrs (max_over_regs_of_instr valu).

Definition max_over_slots_of_funct (valu: slot * Z * typ -> Z) : Z :=
max_over_instrs (max_over_slots_of_instr valu).

Definition int_callee_save (r: mreg) := 1 + index_int_callee_save r.

Definition float_callee_save (r: mreg) := 1 + index_float_callee_save r.

Definition local_slot (s: slot * Z * typ) :=
match s with (Local, ofs, ty) => ofs + typesize ty | _ => 0 end.

Definition outgoing_slot (s: slot * Z * typ) :=
match s with (Outgoing, ofs, ty) => ofs + typesize ty | _ => 0 end.

Definition outgoing_space (i: instruction) :=
match i with Lcall sig _ => size_arguments sig | _ => 0 end.

Lemma max_over_list_pos:
forall (A: Type) (valu: A -> Z) (l: list A),
max_over_list valu l >= 0.
Proof.
intros until valu. unfold max_over_list.
assert (forall l z, fold_left (fun x y => Zmax x (valu y)) l z >= z).
induction l; simpl; intros.
omega. apply Zge_trans with (Zmax z (valu a)).
auto. apply Zle_ge. apply Zmax1. auto.
Qed.

Lemma max_over_slots_of_funct_pos:
forall (valu: slot * Z * typ -> Z), max_over_slots_of_funct valu >= 0.
Proof.
intros. unfold max_over_slots_of_funct.
unfold max_over_instrs. apply max_over_list_pos.
Qed.

Lemma max_over_regs_of_funct_pos:
forall (valu: mreg -> Z), max_over_regs_of_funct valu >= 0.
Proof.
intros. unfold max_over_regs_of_funct.
unfold max_over_instrs. apply max_over_list_pos.
Qed.

Program Definition function_bounds :=
mkbounds
(max_over_slots_of_funct local_slot)
(max_over_regs_of_funct int_callee_save)
(max_over_regs_of_funct float_callee_save)
(Zmax (max_over_instrs outgoing_space)
(max_over_slots_of_funct outgoing_slot))
(Zmax f.(fn_stacksize) 0)
(max_over_slots_of_funct_pos local_slot)
(max_over_regs_of_funct_pos int_callee_save)
(max_over_regs_of_funct_pos float_callee_save)
_ _.
Next Obligation.
apply Zle_ge. eapply Zle_trans. 2: apply Zmax2.
apply Zge_le. apply max_over_slots_of_funct_pos.
Qed.
Next Obligation.
apply Zle_ge. apply Zmax2.
Qed.

We now show the correctness of the inferred bounds.

Lemma max_over_list_bound:
forall (A: Type) (valu: A -> Z) (l: list A) (x: A),
In x l -> valu x <= max_over_list valu l.
Proof.
intros until x. unfold max_over_list.
assert (forall c z,
let f := fold_left (fun x y => Zmax x (valu y)) c z in
z <= f /\ (In x c -> valu x <= f)).
induction c; simpl; intros.
split. omega. tauto.
elim (IHc (Zmax z (valu a))); intros.
split. apply Zle_trans with (Zmax z (valu a)). apply Zmax1. auto.
intro H1; elim H1; intro.
subst a. apply Zle_trans with (Zmax z (valu x)).
apply Zmax2. auto. auto.
intro. elim (H l 0); intros. auto.
Qed.

Lemma max_over_instrs_bound:
forall (valu: instruction -> Z) i,
In i f.(fn_code) -> valu i <= max_over_instrs valu.
Proof.
intros. unfold max_over_instrs. apply max_over_list_bound; auto.
Qed.

Lemma max_over_regs_of_funct_bound:
forall (valu: mreg -> Z) i r,
In i f.(fn_code) -> In r (regs_of_instr i) ->
valu r <= max_over_regs_of_funct valu.
Proof.
intros. unfold max_over_regs_of_funct.
apply Zle_trans with (max_over_regs_of_instr valu i).
unfold max_over_regs_of_instr. apply max_over_list_bound. auto.
apply max_over_instrs_bound. auto.
Qed.

Lemma max_over_slots_of_funct_bound:
forall (valu: slot * Z * typ -> Z) i s,
In i f.(fn_code) -> In s (slots_of_instr i) ->
valu s <= max_over_slots_of_funct valu.
Proof.
intros. unfold max_over_slots_of_funct.
apply Zle_trans with (max_over_slots_of_instr valu i).
unfold max_over_slots_of_instr. apply max_over_list_bound. auto.
apply max_over_instrs_bound. auto.
Qed.

Lemma int_callee_save_bound:
forall i r,
In i f.(fn_code) -> In r (regs_of_instr i) ->
index_int_callee_save r < bound_int_callee_save function_bounds.
Proof.
intros. apply Zlt_le_trans with (int_callee_save r).
unfold int_callee_save. omega.
unfold function_bounds, bound_int_callee_save.
eapply max_over_regs_of_funct_bound; eauto.
Qed.

Lemma float_callee_save_bound:
forall i r,
In i f.(fn_code) -> In r (regs_of_instr i) ->
index_float_callee_save r < bound_float_callee_save function_bounds.
Proof.
intros. apply Zlt_le_trans with (float_callee_save r).
unfold float_callee_save. omega.
unfold function_bounds, bound_float_callee_save.
eapply max_over_regs_of_funct_bound; eauto.
Qed.

Lemma local_slot_bound:
forall i ofs ty,
In i f.(fn_code) -> In (Local, ofs, ty) (slots_of_instr i) ->
ofs + typesize ty <= bound_local function_bounds.
Proof.
intros.
unfold function_bounds, bound_local.
change (ofs + typesize ty) with (local_slot (Local, ofs, ty)).
eapply max_over_slots_of_funct_bound; eauto.
Qed.

Lemma outgoing_slot_bound:
forall i ofs ty,
In i f.(fn_code) -> In (Outgoing, ofs, ty) (slots_of_instr i) ->
ofs + typesize ty <= bound_outgoing function_bounds.
Proof.
intros. change (ofs + typesize ty) with (outgoing_slot (Outgoing, ofs, ty)).
unfold function_bounds, bound_outgoing.
apply Zmax_bound_r. eapply max_over_slots_of_funct_bound; eauto.
Qed.

Lemma size_arguments_bound:
forall sig ros,
In (Lcall sig ros) f.(fn_code) ->
size_arguments sig <= bound_outgoing function_bounds.
Proof.
intros. change (size_arguments sig) with (outgoing_space (Lcall sig ros)).
unfold function_bounds, bound_outgoing.
apply Zmax_bound_l. apply max_over_instrs_bound; auto.
Qed.

Consequently, all machine registers or stack slots mentioned by one of the instructions of function f are within bounds.

Lemma mreg_is_within_bounds:
forall i, In i f.(fn_code) ->
forall r, In r (regs_of_instr i) ->
mreg_within_bounds function_bounds r.
Proof.
intros. unfold mreg_within_bounds. split.
eapply int_callee_save_bound; eauto.
eapply float_callee_save_bound; eauto.
Qed.

Lemma slot_is_within_bounds:
forall i, In i f.(fn_code) ->
forall sl ty ofs, In (sl, ofs, ty) (slots_of_instr i) ->
slot_within_bounds function_bounds sl ofs ty.
Proof.
intros. unfold slot_within_bounds.
destruct sl.
eapply local_slot_bound; eauto.
auto.
eapply outgoing_slot_bound; eauto.
Qed.

Lemma slots_of_locs_charact:
forall sl ofs ty l, In (sl, ofs, ty) (slots_of_locs l) <-> In (S sl ofs ty) l.
Proof.
induction l; simpl; intros.
tauto.
destruct a; simpl; intuition congruence.
Qed.

It follows that every instruction in the function is within bounds, in the sense of the instr_within_bounds predicate.

Lemma instr_is_within_bounds:
forall i,
In i f.(fn_code) ->
instr_within_bounds function_bounds i.
Proof.
intros;
destruct i;
generalize (mreg_is_within_bounds _ H); generalize (slot_is_within_bounds _ H);
simpl; intros; auto.
call *)  eapply size_arguments_bound; eauto.
builtin *)  split; intros.
apply H1. apply in_or_app; auto.
apply H0. rewrite slots_of_locs_charact; auto.
Qed.

Lemma function_is_within_bounds:
function_within_bounds f function_bounds.
Proof.
intros; red; intros. apply instr_is_within_bounds; auto.
Qed.

End BOUNDS.